Microrheology uses colloidal probes to obtain rheological properties (e.g. viscosity, elasticity or viscoelasticity) of soft materials. The small probe size is advantageous in that it requires small sample volumes, allows incorporation of probes into materials during normal processing, and enables measurement of local properties of heterogeneous samples.

Passive microrheology relies on thermal fluctuations to drive probe motion. By its nature, passive microrheology is limited to materials soft enough to respond to thermal forcing, and to only the linear response properties of such materials. Active microrheology circumvents these limitations by applying an external force to drive probe motion. While it is more involved experimentally, it enables one to investigate the rheology of stiffer materials, as well as holds potential to probe nonlinear rheological properties (e.g. shear thinning, yield stress, and normal stress differences). However, the strong forcing drives the material out of equilibrium, and renders the theoretical foundation of passive microrheology invalid. This leaves no clear theoretical path to relate the probe motion in nonlinear microrheology to the nonlinear rheological properties.

The work presented in this dissertation seeks to develop a theoretical foundation for active, nonlinear microrheology. We use computational techniques to investigate the relationship between the non-equilibrium microstructure and nonlinear rheological properties of soft materials.

We first show that gentle motion of a microrheological probe drives stresses within the bulk material, whose retarding influence on the probe is in quantitative agreement with macroscopic linear shear rheology. Guided by this, we then examine the bulk microstructural stresses established around a strongly forced microrheological probe. In the nonlinear regime of active microrheology, a variety of new features arise -- non-viscometric flows, spatially inhomogeneous strains, and Lagrangian unsteadiness -- that potentially complicate the theoretical interpretation of these stresses and their relation to macroscopic rheological properties. We investigate and quantify each of these possible issues, and propose alternative probe shapes and probe motions to mitigate these effects, and illustrate conditions under which macroscopic rheological properties may be extracted from nonlinear microrheology measurements.